The paper is organized as follows. Since the subject of the article is rather technical, we partition it into the small sections distinguishing the major successive steps of the derivation, which culminates in Sec.  where the results and its implications are discussed. In Sec. , we provide general formulas for the one-loop correction to the free energy. The main technical tool, which we shall employ to derive the high-temperature expansion, is the heat kernel of a Laplacian operator (see [1]} for review). Therefore, in Secs. , , and  we reduce our problem to the evaluation of the heat kernel expansion coefficients. At first, Sec. , we reduce the three-dimensional problem to the four-dimensional one to provide the explicit general covariance to the expansion. The method, which is used here, is a finite dimensional analog of the gauge fixing procedure in the functional integral. Then, in Sec. , we resum the heat kernel expansion applying the theorem proved in [2]}. Notice that we do not use the conformal transformation to derive the high-temperature expansion [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, but apply the method directly to the Fourier transformed Klein-Gordon operator. So, our approach is rather close to the one used in [11]}. After that, the problem becomes in essence the same as in a flat spacetime. In Sec. , we evaluate the expansions of necessary integrals employing the procedure used in [12]}. Then, in Sec. , a general formula for the high-temperature expansion of the free energy is obtained. In Sec. , we particularize the general formulas to the three-dimensional space and single out the divergent and finite parts of the high-temperature expansion. At this point we essentially employ the results of [13]}, where the heat kernel coefficients were derived up to \(a_6\) . In Appendix , we give some useful relations for the metric possessing the Killing vector. In Appendix , the relevant parts of the heat kernel coefficients borrowed from [1]}, [13]}, [16]}, [17]} are calculated.
